Infinite Geometric Series

An infinite
geometric series
is the sum of an infinite
geometric sequence
. This series would have no last term. The general form of the infinite geometric series is
a
1
+
a
1
r
+
a
1
r
2
+
a
1
r
3
+
...
, where
a
1
is the first term and
r
is the common ratio.

We can find the sum of all finite geometric series. But in the case of an infinite geometric series when the
common ratio
is greater than one, the terms in the sequence will get larger and larger and if you add the larger numbers, you won't get a final answer. The only possible answer would be infinity. So, we don't deal with the common ratio greater than one for an infinite geometric series.

If the common ratio
r
lies between
−
1
to
1
, we can have the sum of an infinite geometric series. That is, the sum exits for
|
r
|
<
1
.

The sum
S
of an infinite geometric series with
−
1
<
r
<
1
is given by the formula,

S
=
a
1
1
−
r

An infinite series that has a sum is called a convergent series and the sum
S
n
is called the partial sum of the series.